Figure 1

Perhaps you've learned from a calculus class that
as you roll a circular disk along a straight line,
that the area under the cycloid swept out by
following a point on the edge of the disk between
two successive points of tangency is exactly 3 times
the area of the disk.
But did you know that a very similar
fact is true for polygons?
For instance, take a square on a flat line, and mark
one corner on the line
with a red dot. Now "roll" it along the
line by pivoting the square around the corner
that touches the line. Each time it comes to a rest,
mark the position of the red dot. When the red dot
again touches the line, stop.
Connect the red dots with straight lines.
(These are dotted lines in the Figure.) The area
under this polygonal region will be 3 times the area of
the square. You can verify this in Figure 1.
The same holds for pentagons, hexagons, and any regular
ngon!
Presentation Suggestions:
Draw examples on the board! Challenge students to
show this fact true for a triangle or
a pentagon (harder).
The Math Behind the Fact:
Regular ngons with a large number of sides are
approximately circular, and the polygonal path obtained
by connecting the dots will approximately converge to
the path taken by a point on the edge of the disk!
This recovers the result for the cycloid.
How to Cite this Page:
Su, Francis E., et al. "Rolling Polygons."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
